Quantifying noisy attractors: from heteroclinic to excitable networks

This is the author accepted manuscript. The final version is available from the Society for Industrial and Applied Mathematics via the DOI in this record.Attractors of dynamical systems may be networks in phase space that can be heteroclinic (where there are dynamical connections between simple invariant sets) or excitable (where a perturbation threshold needs to be crossed to a dynamical connection between “nodes”). Such network attractors can display a high degree of sensitivity to noise both in terms of the regions of phase space visited and in terms of the sequence of transitions around the network. The two types of network are intimately related—one can directly bifurcate to the other. In this paper we attempt to quantify the effect of additive noise on such network attractors. Noise increases the average rate at which the networks are explored, and can result in “macroscopic” random motion around the network. We perform an asymptotic analysis of local behaviour of an escape model near heteroclinic/excitable nodes in the limit of noise η → 0 + as a model for the mean residence time T near equilibria. The heteroclinic network case has T proportional to − ln η while the excitable network has T given by a Kramers’ law, proportional to exp(B/η2 ). There is singular scaling behaviour (where T is proportional to 1/η) at the bifurcation between the two types of network. We also explore transition probabilities between nodes of the network in the presence of anisotropic noise. For low levels of noise, numerical results suggest that a (heteroclinic or excitable) network can approximately realise any set of transition probabilities and any sufficiently large mean residence times at the given nodes. We show that this can be well modelled in our example network by multiple independent escape processes, where the direction of first escape determines the transition. This suggests that it is feasible to design noisy network attractors with arbitrary Markov transition probabilities and residence times.We thank many people for stimulating conversations that contributed to the development of this paper: in particular Chris Bick, Nils Berglund, Mike Field, John Terry, Ilze Ziedins. We thank the London Mathematical Society for support of a visit of CMP to Exeter, and the University of Auckland Research Council for supporting a visit of PA to Auckland during the development of this research. PA gratefully acknowledges the financial support of the EPSRC via grant EP/N014391/1

Sensitive finite state computations using a distributed network with a noisy network attractor

This is the author accepted manuscript. The final version is available from IEEE via the DOI in this record.We exhibit a class of smooth continuous-state neural-inspired networks composed of simple nonlinear elements that can be made to function as a finite state computational machine. We give an explicit construction of arbitrary finitestate virtual machines in the spatio-temporal dynamics of the network. The dynamics of the functional network can be completely characterised as a “noisy network attractor” in phase space operating in either an “excitable” or a “free-running” regime, respectively corresponding to excitable or heteroclinic connections between states. The regime depends on the sign of an “excitability parameter”. Viewing the network as a nonlinear stochastic differential equation where deterministic (signal) and/or stochastic (noise) input are applied to any element, we explore the influence of signal to noise ratio on the error rate of the computations. The free-running regime is extremely sensitive to inputs: arbitrarily small amplitude perturbations can be used to perform computations with the system as long as the input dominates the noise. We find a counter-intuitive regime where increasing noise amplitude can lead to more, rather than less, accurate computation. We suggest that noisy network attractors will be useful for understanding neural networks that reliably and sensitively perform finite-state computations in a noisy environment.PA gratefully acknowledges the financial support of the EPSRC via grant EP/N014391/1. CMP acknowledges travel funding from the University of Auckland and support from the London Mathematical Laboratory

Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control

For many years it was believed that an unstable periodic orbit with an odd number of real Floquet multipliers greater than unity cannot be stabilized by the time-delayed feedback control mechanism of Pyragus. A recent paper by Fiedler et al uses the normal form of a subcritical Hopf bifurcation to give a counterexample to this theorem. Using the Lorenz equations as an example, we demonstrate that the stabilization mechanism identified by Fiedler et al for the Hopf normal form can also apply to unstable periodic orbits created by subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our analysis focuses on a particular codimension-two bifurcation that captures the stabilization mechanism in the Hopf normal form example, and we show that the same codimension-two bifurcation is present in the Lorenz equations with appropriately chosen Pyragus-type time-delayed feedback. This example suggests a possible strategy for choosing the feedback gain matrix in Pyragus control of unstable periodic orbits that arise from a subcritical Hopf bifurcation of a stable equilibrium. In particular, our choice of feedback gain matrix is informed by the Fiedler et al example, and it works over a broad range of parameters, despite the fact that a center-manifold reduction of the higher-dimensional problem does not lead to their model problem.Comment: 21 pages, 8 figures, to appear in PR

Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition

Dynamical systems containing heteroclinic cycles and networks can be invoked as models of intransitive competition between three or more species. When populations are assumed to be well-mixed, a system of ordinary differential equations (ODEs) describes the interaction model. Spatially extending these equations with diffusion terms creates a system of partial differential equations which captures both the spatial distribution and mobility of species. In one spatial dimension, travelling wave solutions can be observed, which correspond to periodic orbits in ODEs that describe the system in a steady-state travelling frame of reference. These new ODEs also contain a heteroclinic structure. For three species in cyclic competition, the topology of the heteroclinic cycle in the well-mixed model is preserved in the steady-state travelling frame of reference. We demonstrate that with four species, the heteroclinic cycle which exists in the well-mixed system becomes a heteroclinic network in the travelling frame of reference, with additional heteroclinic orbits connecting equilibria not connected in the original cycle. We find new types of travelling waves which are created in symmetry-breaking bifurcations and destroyed in an orbit flip bifurcation with a cycle between only two species. These new cycles explain the existence of "defensive alliances" observed in previous numerical experiments. We further describe the structure of the heteroclitic network for any number of species, and we conjecture how these results may generalise to systems of any arbitrary number of species in cyclic competition

ANTIBODY RESPONSES TO ANTIGENIC DETERMINANTS OF INFLUENZA VIRUS HEMAGGLUTININ : I. Thymus Dependence of Antibody Formation and Thymus Independence of Immunological Memory

Using immunodiffusion methods it has been shown that purified hemagglutinin (HA) extracted from two related strains of influenza A viruses (A/PR8/34 and A/FM1/47) have two distinct antigenic determinants, or groups of determinants. One determinant is cross-reactive while the other is strain-specific. Antisera raised in normal mice against HA were shown to contain two populations of antibody molecules, each directed against one of the determinants. Immunization of thymus-deprived (TXBM) mice showed a strong thymus dependence of antibody formation to HA. However, the thymus dependence of antibody formation against the cross-reactive determinant could be overcome by repeated inoculations of HA in TXBM mice, indicating a different handling of two portions of the same molecule by the immunological system. Strong, secondary-type responses to the strain-specific determinant were observed in primed thymus-deprived mice after reconstitution with virgin thymus cells, showing that specific immunological memory was elicited by this determinant despite the absence of detectable antibody secretion. These findings are interpreted as examples of immunological recognition and memory mediated by B lymphocytes and discussed in terms of mechanisms of T and B lymphocyte co-operation. It is suggested that the helper effect of T lymphocytes is exerted at a late stage in the differentiation of specific populations of B cells into antibody-secreting cells

Sprint interval training (SIT) reduces serum epidermal growth factor (EGF), but not other inflammatory cytokines in trained older men

Purpose The present study aimed to investigate the effect of age on circulating pro- and anti-inflammatory cytokines and growth factors. A secondary aim was to investigate whether a novel sprint interval training (SIT) intervention (3 × 20 s ‘all out’ static sprints, twice a week for 8 weeks) would affect inflammatory markers in older men. Methods Nine older men [68 (1) years] and eleven younger men [28 (2) years] comprised the younger group. Aerobic fitness and inflammatory markers were taken at baseline for both groups and following the SIT intervention for the older group. Results Interleukin (IL)-8, vascular endothelial growth factor (VEGF), and monocyte chemoattractant protein-1 (MCP-1) were unchanged for the older and younger groups at baseline (IL-8, p = 0.819; MCP-1, p = 0.248; VEGF, p = 0.264). Epidermal growth factor (EGF) was greater in the older group compared to the younger group at baseline [142 (20) pg mL−1 and 60 (12) pg mL−1, respectively, p = 0.001, Cohen's d = 1.64]. Following SIT, older men decreased EGF to 100 (12) pg mL−1 which was similar to that of young men who did not undergo training (p = 0.113, Cohen's d = 1.07). Conclusion Older aerobically trained men have greater serum EGF than younger aerobically trained men. A novel SIT intervention in older men can shift circulating EGF towards trained younger concentrations. As lower EGF has previously been associated with longevity in C. elegans, the manipulative effect of SIT on EGF in healthy ageing in the human may be of further interest

Time-delayed feedback control in astrodynamics

In this paper we present time-delayed feedback control (TDFC) for the purpose of autonomously driving trajectories of nonlinear systems into periodic orbits. As the generation of periodic orbits is a major component of many problems in astodynamics we propose this method as a useful tool in such applications. To motivate the use of this method we apply it to a number of well known problems in the astrodynamics literature. Firstly, TDFC is applied to control in the chaotic attitude motion of an asymmetric satellite in an elliptical orbit. Secondly, we apply TDFC to the problem of maintaining a spacecraft in a periodic orbit about a body with large ellipticity (such as an asteroid) and finally, we apply TDFC to eliminate the drift between two satellites in low Earth orbits to ensure their relative motion is bounded

The echo index and multistability in input-driven recurrent neural networks

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.A recurrent neural network (RNN) possesses the echo state property (ESP) if, for a given input sequence, it “forgets” any internal states of the driven (nonautonomous) system and asymptotically follows a unique, possibly complex trajectory. The lack of ESP is conventionally understood as a lack of reliable behaviour in RNNs. Here, we show that RNNs can reliably perform computations under a more general principle that accounts only for their local behaviour in phase space. To this end, we formulate a generalisation of the ESP and introduce an echo index to characterise the number of simultaneously stable responses of a driven RNN. We show that it is possible for the echo index to change with inputs, highlighting a potential source of computational errors in RNNs due to characteristics of the inputs driving the dynamics.Engineering and Physical Sciences Research Council (EPSRC)Canada Research Chairs programNZ Marsden fun

Classification and stability of simple homoclinic cycles in R^5

The paper presents a complete study of simple homoclinic cycles in R^5. We find all symmetry groups Gamma such that a Gamma-equivariant dynamical system in R^5 can possess a simple homoclinic cycle. We introduce a classification of simple homoclinic cycles in R^n based on the action of the system symmetry group. For systems in R^5, we list all classes of simple homoclinic cycles. For each class, we derive necessary and sufficient conditions for asymptotic stability and fragmentary asymptotic stability in terms of eigenvalues of linearisation near the steady state involved in the cycle. For any action of the groups Gamma which can give rise to a simple homoclinic cycle, we list classes to which the respective homoclinic cycles belong, thus determining conditions for asymptotic stability of these cycles.Comment: 34 pp., 4 tables, 30 references. Submitted to Nonlinearit